在复平面处处连续
可导的充分条件:
解析函数
所有解析点的合集必为开集,不可能仅在一个点或一条曲线上解析
平面上处处解析的函数
处处不解析的函数
:参数方程求解
调和函数
Example
复积分
非闭区间
参数方程法
Example , to 直线段
Example , C:上半单位圆周线,起点-1
牛顿-莱布尼兹公式
周线积分
参数方程
Example
推广的柯西积分定理
柯西积分公式
Example
留数定理
f(z)在 去心邻域的洛朗展开中 系数
本性奇点
可去奇点
极点
留数定理应用
泰勒展开
洛朗展开
傅里叶变换
拉普拉斯变换
add lim
if is partial derivative at , , are continuous at , and is differentiable at .
Example 1 .
Example 2
analytic function
Example
are not analytic
Construct: Given a differentiable function u,(v), if exsist a v,(u) make analytical
C-R
C-R
Harmonic Function : ( ) , continuous
Example:
analytic u,v harmonic in D and support C-R Equation.
If and harmonic in D and satisfying C-R equation , , then v is the convergent harmonic function of u.
Partial Integrate Method
Example
Primary Analytic Function
example
Example
Example
Example
, ( )
*
Example: solve
zero point of is
zero point of is
Example
Example
Homework 2.9(1)(4), 2.13(1)(2)(4), 2.14
Complex Integral
If continuous on smooth curve from to ( , are continuous),
Parametric Equation
(Important!) Parametric Equation method: *
Example:
Prove: where
Cauchy Integral Theorm
integral in D is unrelated to path
If is analytic in single connected region , for any contour in .
Example1:
Example2: solve
singular point:
Construct :
Cauchy integral theorm on complex connected region
is analytic in bounded in and (contours) and continuous on , then
Example: solve , bound 1 and -1
Construct , around -1 and 1.
is analytic in , , is analytic in and
is analytic in single connect region ,
Calculate Complex Integration
not close
contour
Cauchy Integral Formula
If is analytic in surrounded by contour , continuous on and , for all ,
Application:
Example:
Example2:
Example3:
if analytic on ,
Example:
Example2:
Example3:
infinite series
convergence on
convergent then and convergent.
Example
convergent, then convergent.
power series
Abel's Theorm
Taylor Series
Taylor Expansion Theorem: If analytic in , for all , if in , could be expand into , where
: Taylor Coefficient
: Expand Center
,
Trick:
Example 1: ,
Taylor Expansion Method
Solve Taylor Coefficient Directly:
Solve with known expansion like
,
,
,
,
,
,
Derivative within contour
Example: ,
Example2: ,
Example3: ,
Laurent Series
Laurent Expansion
Definition: (1) and (2), (1)+(2) called Laurent Series.
Convergence Region:
Let , then
convergent
(2) convergent in
When ,
could be derivative by steps within
could be integrate along within
Laurent Expansion Theorem
Example1: ,
Laurent Expansion Method
Expand through exsisted Taylor Expansion
is an analyic point on
Example ,
Example: ,
Integrate/Derivate by steps within domain region
Example
Integrate
Residue:
Residue Theroem:
Isolated Singularity
Classification of Isolated Singularity
Removable Singularity
Main part of equals to 0, ,
Append definition then analytic on .
Example 1: Prove that is the removable singularity of
Th: is the removable singularity of
L'Hospital Rule:
,
,
, is removable singularity
Pole Singularity
and , then is the m-order pole singularity of
Example: prove that is the 2018-order polar singularity of
methods
,
Example:
is the pole singularity of can be expressed as and analytic in and
Requirements
Example: ,
is m-order pole singularity of is m-order 0-point of
Example: ,
Example 2:
or
where is removable singularity and is m-order pole singuarity, then is m-order pole singularity
Essential Singularity
Residue of removable singularity:
Residue of essential singularity: solve with Laurent Expansion
Example:
Example:
Residue of m-order pole singularity
solve with Laurent Expansion:
Example:
if is m-order pole singularity of ,
Especially when ,
Example1
: 2-order pole singularity
: 1-order
Example
Example
Example
Process
Application of Residue theorem
let
Example:
where and
then , where and is the singularity on above part of
Example:
Example2:
Fourier Transform
Dirichlet Requiremnt:
using Euler Equation
Fourier Integration Exisit theorem
LIST:
Function (Dirac Function)
Fourier Transformation of function
Example
Example
Example
Characteristics
Linear
Translate
Example
Example
Example
Similarity
differential
If ,then
Generally
Example
integral
Convolution
Preparation For Laplace Transform
Example
Example
Example
Example
Laplace Transformation
Definition of Laplace Transformation
Examples:
solve
Solve
Mini-
2 characteristics
Linear
Example 1:
Example 2:
Translate
Example 1:
Example 2:
unknown
Derivative
Assume that , then
Generally,
Example
Example
Example
Example
Differential
Assume that , then
Especially when ,
Example Important
Example
Integral
Assume that , then
, where
Example
Example
Example
Example
Example
Examples: